Two scalar fields of the N=4 SYM theory: Long local operators: Can be mapped to the spin chain states: The mixing matrix is an integrable spin chain Hamiltonian! Minahan, Zarembo

sl(2) sector: Can be diagonalized by BAE

In scaling limitthe Bethe roots condense into cuts Cuts of roots correspond to the classical solutions

Expanding Bathe ansatz equation for sl(2) spin chain we will find Korchemsky; Kazakov; Beisert, Tseytlin, Zarembo where Then the BAE becomes to the 1/L order

Korchemsky; N.G. V Kazakov BAE is equivalent to the absence of poles at u=u j Baxter “polynomial” Let us define q(x) by the following equation exp(i q(x)) is a double valued function Expanding T(u) We get for q

BAE for SU(1,2) spin chaine Where

N.G. P. Vieira For su(2,1) spin chain there are several Baxter polynomials We can define some algebraic curve by the polynomial equation Then for each branch cut we must have

Expanding in L we get Where and On C 23 On C 13

Taking into account this mismatch we can write equation for density Bethe roots form bound states, but they are separated by  1

Beisert, Staudacher; Beisert,Eden,Staudacher

For general configuration of roots we have the following equation Where

From “stack” to “zipper”

Bosonic duality